Equivariant Verlinde Formula from Fivebranes and Vortices

We study complex Chern-Simons theory on a Seifert manifold M (3) by embedding it into string theory. We show that complex Chern-Simons theory on M (3) is equivalent to a topologically twisted supersymmetric theory and its partition function can be naturally regularized by turning on a mass parameter. We find that the dimensional reduction of this theory to 2d gives the low energy dynamics of vortices in four-dimensional gauge theory, the fact apparently overlooked in the vortex literature. We also generalize the relations between (1) the Verlinde algebra, (2) quantum cohomology of the Grassmannian, (3) Chern-Simons theory on and (4) index of a spin (c) Dirac operator on the moduli space of flat connections to a new set of relations between (1) the equivariant Verlinde algebra for a complex group, (2) the equivariant quantum K-theory of the vortex moduli space, (3) complex Chern-Simons theory on and (4) the equivariant index of a spin (c) Dirac operator on the moduli space of Higgs bundles.